Behaviour of Quantum Chromodynamics Near an Infrared Fixed Point Galib O. M. Souadi Special Research Centre for the Subatomic Structure of Matter School of Chemistry and Physics A Thesis Submitted for the Degree of Doctor of Philosophy April 2014 Abstract In perturbation theory, the QCD running coupling depends on the renormalization scheme or is parameterised by a physical process. The problem is that artefacts of this ambiguity may upset physical conclusions outside the asymptotically free region, in particular near an infrared fixed point. Thus, a non-perturbative definition for the QCD running coupling is required that should be a mono- tonic and analytic function of the space-like energy scale Q2. The most physical coupling is Grunberg's definition for the running 2 coupling as an effective charge αG(Q ). However, we find that it works only for sufficiently high energy scales. At some finite val- ues of the energy scale, near the top of the resonance region, the 2 β-function associated with αG(Q ) has a false zero below which 2 αG(Q ) decreases. We test this conclusion further by applying chiral perturbation theory to the running coupling based on the method of effec- tive charges. We consider the Drell-Yan ratio R(q2) and Adler- function D(Q2) in the time-like and space-like domains, respec- tively. These quantities tend to a finite value in the both in- frared and ultraviolet limits: R(0) = D(0) = 0:5 for π±;K± and R(1) = D(1) = 2 for the light quarks. This means that the running coupling becomes negative in the infrared limit. There- fore, neither the time-like nor the space-like effective charges is a monotonic function over the whole energy scale. We also try to cancel the space-like pole of the proposed exact β-function, using the property of renormalization scheme depen- dence. Our modified exact β-function is non-singular for all finite values of the running coupling α and has an infrared fixed point even for a small number of quark flavours. Statement of Originality I certify that this work contains no material which has been ac- cepted for the award of any other degree or diploma in my name, in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously pub- lished or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission in my name, for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Ade- laide and where applicable, any partner institution responsible for the joint-award of this degree. I give consent to this copy of my thesis, when deposited in the Uni- versity Library, being made available for loan and photocopying, subject to the provisions of the Copyright Act 1968. I also give permission for the digital version of my thesis to be made available on the web, via the University's digital research repository, the Library Search and also through web search en- gines, unless permission has been granted by the University to restrict access for a period of time. Galib O. SOUADI List of workshop proceeding and future publication based on this thesis: • G. O. Souadi, When is an Infrared Fixed Point of QCD Phys- ical?, poster presented at workshop "Determination of the Fundamental Parameters of QCD", Nanyang Technological University, Singapore, March 18-21, 2013. • R. J. Crewther and G. O. Souadi, When is a QCD Infrared Fixed Point Physical?, in preparation. Acknowledgements The completion of any work is too hard without the contribution of other people. During my PhD, I had a support from many people, whose should have a great acknowledgement from me. First and foremost, I have to thank my supervisors for their sup- port, time, great ideas to complete my PhD. In particular, I would like give a great thank to my principal supervisor Dr. Rod Crewther for his effort, ideas and guidance, wisdom and helpful comments throughout my PhD. The support from him stems from the beginning of my studying at the University of Adelaide for the degree of Master and then PhD. Also, thanks to him for the great comments and feedback during writing this thesis. Furthermore, I would like to thank my second supervisor Dr. Ross Young for useful comments. I thank all staff of CSSM and School of Chemistry and Physics for administering and providing technical support. I owe deep gratitude to the government of Saudi Arabia for the financial support to complete my PhD, in particular King Khalid University. Thanks to Saudi Arabia Culture Mission (SACM) in Australia for the excellent communication with my sponsor in Saudi Arabia. To Department of Physics at King Khalid Uni- versity for choosing me to complete my postgraduate education, Master and then PhD at the University of Adelaide. In particular, I am grateful to Profs. Abdulaziz AlShahrani , Ali AlHajry and Ali AlKamli for giving me this opportunity. I also would like to thank my friends at Adelaide University who are help me during this work. In particular, I thank Lewis Tun- stall for his helpful discussions and comments on my PhD project. Also, thanks to him for pointing me to many important papers related to my work. Further, I would like to thank my friend Ali Alkathiri for providing a technical help at some stages of writing my thesis. Also, thanks to other friends outside of physics commu- nity for their moral support. I also have to thank my nephew A. AlFraji and Elite Editing Centre for proofreading at some stages of my thesis. Last but not least, I would like to send a special thank to all members of my family for their encouragement to complete my postgraduate education, in particular my mother for her love and prayer. The biggest thank is to my wife for her love and support to complete my PhD. She shared me all problems that I had met during my work and always encouraged me to pass these problems. Thanks to her very much. To my children, Yazad, Ziyad, Ayan and Ellen, who are made my world so beautiful. Contents Contents vi List of Figures ix List of Tables xiii 1 Introduction1 2 Quantum Chromodynamics7 2.1 QCD Lagrangian . .7 2.1.1 Gauge Invariance . .7 2.1.2 QCD Quantisation . 10 2.2 Renormalization Procedure . 12 2.3 Perturbative QCD . 15 2.3.1 The QCD β-Function . 15 2.3.2 Asymptotic Freedom . 16 2.3.3 Annihilation of e+e− into Hadrons . 20 2.4 Non-Perturbative QCD . 23 2.4.1 Effective Field Theories . 23 2.4.2 Proposed Behaviour for the QCD β-function at low- energy scales . 25 3 Perturbative Running Coupling and Renormalization Scheme 28 3.1 Perturbative Running Coupling . 28 3.2 The Problem of Renormalization Scheme . 31 3.3 Proposed Solutions for the Problem . 33 3.3.1 The 't Hooft Scheme . 34 vi CONTENTS 3.3.2 Effective Charges Method . 37 3.3.2.1 The Gell-Mann-Low -Function . 37 3.3.2.2 The Generalisation of the GML -Function into QCD . 39 3.4 Physical Running Coupling . 41 4 Analysis of Perturbative and Non-Perturbative Infrared Fixed Point 43 4.1 Perturbative Infrared Fixed Point . 43 4.1.1 The Banks-Zaks Infrared Fixed Point . 44 4.1.2 The Effect of Higher-Loop Orders . 46 4.1.3 Large Limits: Nf ;Nc ! 1 ................ 50 4.2 Non-Perturbative Infrared Fixed Point . 51 4.2.1 Schr¨odinger Functional Analysis . 52 4.2.2 Dyson-Schwinger Equations Analysis . 54 4.2.3 Anti-de Sitter Analysis . 58 5 The Infrared Behaviour of Proposed Exact β-Functions 62 5.1 Supersymmetric β-Function . 63 5.2 Ryttov-Sannino (RS) β-Function . 65 5.2.1 The Singularity of the βRS-Function . 65 5.2.2 Non-trivial Fixed Point . 67 5.2.3 Comparison of the Banks-Zaks and Ryttov-Sannino Fixed Points . 69 5.3 Modified βRS-Function . 72 5.3.1 Renormalization Gauge Transformation . 72 5.3.2 Infrared Fixed point . 73 6 Chiral Analysis of QCD at Low-Energy Scales 79 6.1 Chiral Perturbation Theory . 80 6.1.1 Chiral Symmetry . 80 6.1.2 Effective Lagrangians for Strong Interactions . 82 6.1.2.1 Transformation Properties of the Goldstone Bosons . 82 6.1.2.2 The Lowest-Order Effective Lagrangian . 84 vii CONTENTS 6.2 Electromagnetic Interactions . 85 6.2.1 External Fields . 85 6.2.2 Effective Lagrangian for Electromagnetic Interactions . 87 6.3 The Physical Quantity R(q2)................... 90 6.4 QCD Effective Charges . 93 7 Conclusion 96 7.1 Summary of Results . 96 7.2 What Next? . 98 A Feynman Diagrams 100 B Reference Formulaes 102 B.1 Gell-Mann Matrices . 102 B.2 Gamma Matrices . 103 B.3 Loop Integrals . 103 B.4 Dimensional Regularization . 104 C The β-Function and Anomalous Dimension 107 C.1 Perturbative Coefficients . 107 C.2 The SU(Nc) Coloured Gauge Group . 110 D Proposed Exact β-Function at a Small α 112 D.1 Proof 1 . 112 D.2 Proof 2 . 113 D.3 Proof 3 . 115 E Loop Integrals 116 E.1 Meson-Loop Integral . 116 E.2 Quark-Loop Integral . 119 References 121 viii List of Figures 2.1 Contributions of fermions and all gauge boson interaction terms of order α to gluon propagator in QCD. 15 2.2 In QCD, the point at the origin is an UV fixed point, where β < 0.